Problem: $\dfrac{ -i - 6j }{ 8 } = \dfrac{ -5i - k }{ -8 }$ Solve for $i$.
Solution: Notice that the left- and right- denominators are opposite $\dfrac{ -i - 6j }{ {8} } = \dfrac{ -5i - k }{ -{8} }$ So we can multiply both sides by $8$ ${8} \cdot \dfrac{ -i - 6j }{ {8} } = {8} \cdot \dfrac{ -5i - k }{ -{8} }$ $-i - 6j = - \cdot \left( -5i - k \right) $ Distribute the negative sign on the right side. $-i - 6j = 5i + k$ $-{1}i - {6}j = {5}i + {1}k$ Combine $i$ terms on the left. $-{i} - 6j = {5i} + k$ $-{6i} - 6j = 1k$ Move the $j$ term to the right. $-6i - {6j} = 1k$ $-6i = k + {6j}$ Isolate $i$ by dividing both sides by its coefficient. $-{6}i = k + 6j$ $i = \dfrac{ k + 6j }{ -{6} }$ Swap signs so the denominator isn't negative. $i = \dfrac{ -{1}k - {6}j }{ {6} }$